Introduction

Decision problems under uncertainty concern the choice between random payoffs. For a rational agent with a known utility function, one random variable is preferred if it maximizes expected utility. This is easy enough in theory. However, in practice it is often difficult to find an agent's utility function. Therefore it would be most useful to know whether a random variable is the dominant choice because it is preferred by all agents whose utility functions share certain general characteristics.

In this chapter, we introduce two such rankings, known as first- and second-order stochastic dominance. These notions apply to pairs of random variables. They indicate when one random variable ranks higher than the other by specifying a condition which the difference between their distribution functions must satisfy.

Essentially, first-order stochastic dominance is a stochastically larger and second-order stochastic dominance a stochastically less volatile or less risky relationship. While the larger random variable is preferred by all agents who prefer higher realizations, the less volatile random variable is preferred by all agents who also dislike risk. In this sense, stochastic dominance theory gives unanimity rules, provided utility functions share certain properties.

Stochastic dominance theory has a bearing on the old issue whether one can judge a random variable as more risky than another regardless of who is the judge, provided that utility functions belong to a class with certain common properties.